Bull. Korean Math. Soc. 52 (2015), No. 2, pp. 421–438 http://dx.doi.org/10.4134/BKMS.2015.52.2.421

NONLINEAR DIFFERENTIAL INCLUSIONS OF SEMIMONOTONE AND CONDENSING TYPE IN HILBERT SPACES

Hossein Abedi and Ruhollah Jahanipur

Abstract. In this paper, we study the existence of classical and gen- eralized solutions for nonlinear differential inclusions x′(t) ∈ F (t, x(t)) in Hilbert spaces in which the multifunction F on the right-hand side is hemicontinuous and satisfies the semimonotone condition or is condens- ing. Our existence results are obtained via the selection and fixed point methods by reducing the problem to an ordinary differential equation. We first prove the existence theorem in finite dimensional spaces and then we generalize the results to the infinite dimensional separable Hilbert spaces. Then we apply the results to prove the existence of the mild solution for semilinear evolution inclusions. At last, we give an example to illustrate the results obtained in the paper.

1. Introduction Our aim is to study the nonlinear differential inclusion of first order x′(t) ∈ F (t, x(t)), (1.1) x(0) = x0, in a Hilbert spaces in which F may be condensing or semimonotone set-valued map. An approach to investigating the existence of solution for a differential inclusion is to reduce it to a problem for ordinary differential equations. In this setting, we are interested to know under what conditions the solution of the corresponding ordinary differential equation belongs to the right side of the given differential inclusion; in other words, under what conditions a such as f(·, ·) exists so that f(t, x) ∈ F (t, x) for every (t, x) and x′(t)= f(t, x(t)). Thus, fixed point theorems and selection theorems to some of which we point out in Section 2, are the main tools in this method. The corresponding ′ ordinary differential equation x (t) = f(t, x(t)) with x(0) = x0, in case that f : R × Rn → Rn is continuous and Lipschitz (or Locally Lipschitz) with

Received September 28, 2013. 2010 Mathematics Subject Classification. Primary 34A60; Secondary 49K24, 47H05. Key words and phrases. differential inclusions, set-valued integral, semimonotone and hemicontinuous multifunctions, condensing multifunctions.

c 2015 Korean Mathematical Society 421 422 HOSSEINABEDIANDRUHOLLAHJAHANIPUR respect to the second variable, according to the Picard-Lindelof theorem [1], has a unique classical solution (local solution). However, it is well-known that continuity condition alone does not suffice for the problem to have even a local solution in infinite dimensional spaces. In a more general case, the function f is considered to satisfy Caratheodory condition and therefore, solutions of Caratheodory type are obtained. To study the semilinear and nonlinear Cauchy differential equations in Hilbert spaces, we refer to [9, 10, 19, 20, 22]. For differential inclusions, the conditions we impose on set-valued map F are usually a combination of two types: First, regularity of the map F such as var- ious kinds of continuity, semicontinuity and monotonicity condition. Second, geometrical conditions such as compactness, connectedness and convexity of the values of F . Existence of solution for differential inclusions have been studied by many authors in the past half century with different application-directed mo- tivations, including the issues of control and optimization, dynamical systems and even biological sciences [11, 12, 15, 16, 21]. In the earlier works, set-valued maps have been usually considered with convex values [6, 7]. Next, in the case that the images are nonconvex, the existence theorems were also studied by some authors. For example, Bressan [7] proved the existence theorems when the values of set-valued map are completely disconnected subsets of a finite di- mensional space. Also, in [2, 3] differential inclusion with convex and nonconvex values on infinite dimensional Banach spaces have systematically been studied. In the case that set-valued map F is nonexpansive or Lipschitz in Hausdorff metric topology, several inequalities such as the Gronwall inequality give us a lot of useful information about properties of the set of trajectories of solutions [2, 7, 21, 24]. However, nonexpansive and Lipschitz conditions are very strong and are not of practical importance. For example, in optimal control problems, set-valued functions are often defined as F (t, x)= {f(t,x,u): u ∈ U}, where U is a metric space and the single-valued function f is defined on [0,T ] × Rn × U into Rn . Here F need not be Lipschitz even if the function f is. Thus, replacing Lipschitz condition with a weaker one, would be very valuable. When the set-valued map A = −F is maximal monotone on an infinite di- −1 mensional , the resolvent map Jλ = (I +λA) and the Yosida ap- 1 proximation Aλ = λ (I −Jλ) of A are single-valued. Also, limλ→0 Aλ(x) ∈ A(x) for every x ∈ D(Jλ). In this case, like nonlinear differential equations with maximal monotone condition [23], first, a set of approximate solutions xλ ′ are obtained for differential equations xλ(t) = Aλ(xλ(t)). The images of these solutions under Aλ constitute a weakly compact subset of the space ∞ ∞ L (0,T ; H) and finally a Cauchy subsequence {xλn }n=1 is extracted such that ′ ′ 2 xλn converges weakly to x in L (0,T ; H) and xλn converges strongly to x in L2(0,T ; H). Since A is semiclosed, this subsequence is convergent to the unique solution x of (1.1). In this paper, we establish the existence and uniqueness of the classical and generalized solution for monotone-type differential inclusion (1.1). Our NONLINEAR DIFFERENTIAL INCLUSIONS OF SEMIMONOTONE TYPE 423 novelty is mainly in the conditions we will impose on the nonlinearity F and in the method we will use to prove the existence result which is, in fact, based upon continuous and measurable selection theorems and Kakutani’s fixed point theorem. The paper is organized as follows. In Section 2, we provide some definitions and preliminaries that are required in the next sections. We have allocated Sections 3 and 4 to the existence of generalized solution for problem (1.1). In these sections, we aim to present our main theorems and results when the multifunction F is semimonotone hemicontinuous or condensing. Section 5 includes an application of the results of the previous sections to the existence of the mild solution for semilinear differential inclusions. Finally, we give an example.

2. Definitions and preliminaries Let X be a . We assume that P (X) is the family of all non-empty subsets of X and Pc(X) (resp. Pc,v(X), Pk,v(X) and Pb,c,v(X)) is the family of all non-empty closed (resp. closed convex, compact convex and bounded closed convex) subsets of X. A set-valued map F : X → P (Y ) where Y is another Banach space, is called upper semicontinuous (u.s.c) if for every open subset U of Y , the set {x ∈ X : F (x) ⊆ U} is open in X. By Proposi- tion 1.2.5 of [18], upper semicontinuity is equivalent to the following: for each ∞ sequence {xn}n=1 in X such that xn → x as n → ∞ and for every ε> 0, there exists a positive integer N such that

F (xn) ⊆ F (x)+ εB, ∀n ≥ N, where B is the open unit ball in Y . The map F is called lower semicontinuous (l.s.c) if for every open subset U of Y , the set {x ∈ X : F (x) ∩ U 6= ∅} is open in X. By Proposition 1.2.6 of [18], lower semicontinuity is equivalent to the ∞ following: for each sequence {xn}n=1 in X which converges to x and for each ∞ y ∈ F (x), there exists a sequence {yn}n=1 in Y such that yn ∈ F (xn) for all n and yn → y as n → ∞. Moreover, the set-valued map F is called closed when its graph {(x, y): y ∈ F (x)} is closed in the product space X × Y and it is called compact if F (X) is a compact subset of Y . If for each x ∈ X there exists a neighborhood Ux of x such that F (Ux) is relatively compact in Y , then F is called locally compact. In this paper, we apply Proposition 1.2.23 of [18] to prove upper semicontinuity of set-valued maps:

Proposition 2.1. Let F : X → Pc(Y ) be closed and locally compact. Then F is u.s.c. Suppose that T > 0 and I = [0,T ] is an interval on the real line equipped with the σ-field of Lebesgue measurable sets. The set-valued map F : I → P (X) is called measurable if for every open subset U of X, the set {t ∈ I : F (t) ∩ U 6= ∅} is measurable in I. To say that F is measurable is the same as 424 HOSSEINABEDIANDRUHOLLAHJAHANIPUR for every x ∈ X, the map y : I → R defined by y(t)= d(x, F (t)) = inf{d(x, y): y ∈ F (t)} is measurable [18]. A single-valued function f : X → Y is called a selection for the multifunction F : X → P (Y ) provided that f(x) ∈ F (x) for all x ∈ X. According to the Michael selection theorem [18], if X is a metric and Y is a Banach space, then any l.s.c set-valued map F : X → Pc,v(Y ) has a continuous selection. Moreover, ∞ if X is separable, then there exists a sequence of continuous selections {fn}n=1 such that for every x ∈ X, F (x) = {fn(x)}. One may also be interested in measurable selections as we do in this paper. We use the Kuratowski-Ryll Nardzewski selection theorem [18] which says that if X is a separable and the set-valued map F : I → Pc(X) is measurable, then F has a measurable selection. A two-variable multifunction F : I × X → P (Y ) is said to satisfy the Caratheodory condition, if (i) for every x ∈ X, the function t → F (t, x) is measurable; (ii) for a.e. t ∈ I, the function x 7→ F (t, x) is u.s.c. The next theorem which is a parametric version of Michael selection theorem, provides the conditions under which a set-valued map has a selection of Caratheodory type on a separable Banach space. Theorem 2.2 ([18]). Let Y be a separable Banach space and the set-valued map F : I × X → Pc,v(Y ) satisfy the following hypotheses: i) The map (t, x) 7→ F (t, x) is measurable; ii) For a.e. t ∈ I, the map x → F (t, x) is l.s.c. Then F has a Caratheodory selection. In other words, there is a function f : I × X → Y which is measurable in the first and continuous in the second variable and f(t, x) ∈ F (t, x) for all (t, x) ∈ I × X.

If F : I → P (X) is measurable, we denote by SF the family of all measurable p p selections of F and for p ≥ 1, we set SF = L (I,X) ∩ SF . A set-valued map 1 F is called integrable if SF 6= ∅. In that case, the Aumann integral of the set-valued map F on I is defined as

1 F (t)dt = f(t)dt : f ∈ SF . ZI nZI o For the properties of Aumann integral of multifunctions, see e.g. [4] or Chapter 8 of [3]. Next, we assume that H is a real separable Hilbert space. Given a set-valued map F : I × H → P (H), for each continuous function x : I → H, we define p p SF,x = {v ∈ L (I; H): v(t) ∈ F (t, x(t)) for a.e. t ∈ [0,T ]} . In view of the definition of the integral of set-valued maps, for any continuous function x : I → H, we have

1 F (t, x(t))dt = v(t)dt : v ∈ SF,x . ZI nZI o NONLINEAR DIFFERENTIAL INCLUSIONS OF SEMIMONOTONE TYPE 425

Consider the following initial value problem for differential inclusion x′(t) ∈ F (t, x(t)), t ∈ [0,T ], (2.1)  x(0) = x0.

Here, x0 ∈ H and F : [0,T ] × H → P (H) is a given set-valued map. An abso- lutely continuous solution for the above differential inclusion is called classical solution. Along with the above differential inclusion, we consider the integral equation t x(t)= x + v(s)ds, t ∈ [0,T ], (2.2) 0 0  x(0) = x0, R 1 where v ∈ SF,x. If the map v is continuous on [0,T ], then the solution x of integral equation (2.2) is continuously differentiable on [0,T ]. This is called the strong solution for problem (2.1). In general, a continuous solution x for the integral equation (2.2) is called a generalized solution for the problem (2.1). In order to prove the existence of the generalized solution for problem (2.1), we impose the following hypotheses on set-valued map F :

(H1) F is semimonotone with constant M > 0; that is, for every x1, x2 ∈ H, t ∈ [0,T ] and y1 ∈ F (t, x1) and y2 ∈ F (t, x2), we have 2 hx1 − x2,y1 − y2i≤ M kx1 − x2k ; (H2) There is a continuous function h : [0,T ] × [0, ∞) → [0, ∞) such that for each t ∈ [0,T ], the function u → h(t,u) is monotone increasing and kF (t, x)k≤ h(t, kxk) for all t ∈ [0,T ] and x ∈ H; ∞ (H3) The map F is hemicontinuous; i.e., for each sequence {xn}n=1 in H which is convergent to x and each t ∈ [0,T ], if y ∈ F (t, x), then there ∞ exists a sequence {yn}n=1 in H such that for all n ≥ 1, yn ∈ F (t, xn) w and yn → y in H. Remark 2.3. Note that if the set-valued map F is l.s.c with respect to the second variable, then F is hemicontinuous. If the set-valued map F is Caratheodory with closed convex values and sat- isfies hypothesis (H2), then any generalized solution of the differential inclusion t (2.1) is classical solution. Indeed, since x(t) − x(s) ∈ s F (τ, x(τ))dτ for all s,t ∈ [0,T ], we have R t t kx(t) − x(s)k≤ kF (τ, x(τ))k dτ ≤ h(τ, kx(τ)k)dτ ≤ K |t − s| Zs Zs for some constant K > 0. Therefore, x is a.e. differentiable. For every ε > 0, there is a positive number δ such that if |t − s| < δ, then F (s, x(s)) ⊆ F (t, x(t)) + εB in which B is the unit open ball in H. So, t x(t) − x(s) ∈ (F (t, x(t)) + εB) dτ = |t − s| (F (t, x(t)) + εB) . Zs 426 HOSSEINABEDIANDRUHOLLAHJAHANIPUR

Therefore, letting ε approach to zero, we obtain x′(t) ∈ F (t, x(t)). Thus x is a classical solution. The Hausdorff measure of noncompactness on the Hilbert space H is a func- tion α : Pb(H) → [0, ∞) which is defined by

n α(A) = inf ε> 0 : A ⊆ Xi, diam(Xi) ≤ ε , ∀A ∈ Pb(H). n i[=1 o

It is not difficult to see that for all A, B ∈ Pb(H), (1) if A ⊆ B, then α(A) ≤ α(B); (2) α(A ∪ B) = max{α(A), α(B)}; (3) α(A + B) ≤ α(A)+ α(B); (4) for every λ ∈ R, α(λA) = |λ| α(A); (5) α(co(A)) = α(A), α(A¯)= α(A), where co(A) is the convex hull of A; and finally (6) α(A) = 0 if and only if A is relatively compact. In general, any function on Pb(H) with the properties (1), (2), (3), (5) and (6), is called a regular measure of noncompactness on H. Let Y be a Banach space and β a regular measure of noncompactness on Y . A set-valued map G : Y → P (Y ) is called β-(countable) condensing if for every (countable) bounded subset A of Y such that β(A) > 0, we have β(G(A)) < β(A). In Section 3, we will need a generalized form of the Monch fixed point theorem (Theorem 4.16 of [1]) for condensing set-valued maps:

Theorem 2.4. Suppose that C is a closed convex subset of the Banach space Y and let the closed set-valued map G : C → Pk,v(C) be condensing with respect to the regular measure of noncompactness β. Then G has a fixed point.

To define the measure of noncompactness for the bounded subsets of the space C(I; H) of all continuous functions x : I → H equipped with the sup- −Lt norm, we set αC (Ω) = supt∈I e α(Ω(t)), for each Ω ∈ Pb(C(I; H)) where L is a suitable constant to be determined latter in the paper and Ω(t) = {x(t): x ∈ Ω}. One can easily check that αC has all properties of the measure of noncompactness except for the regularity property. In other words, if Ω ⊆ C(I; H) is a relatively compact set, then αC (Ω) = 0, but the reverse may not be hold. However, if Ω is a family of equicontinuous functions and αC (Ω) = 0, then Ω is relatively compact subset of C(I; H). To measure the equicontinuity of the subsets of C(I; H), we use the modulus of equicontinuity. This is defined as

modC (Ω) = lim sup sup kx(t1) − x(t2)k . δ→0 ∈ x Ω |t1−t2|≤δ

Note that mod C (Ω) = 0 if and only if Ω ⊆ C(I; H) is an equicontinuous family. Now, we are ready to define a regular measure of noncompactness on C(I; H):

χ(Ω)= max {αC (D), modC (D)} D∈Λ(Ω) for all Ω ∈ Pb(C(I; H)). Here Λ(Ω) is the family of all countable subsets of Ω. NONLINEAR DIFFERENTIAL INCLUSIONS OF SEMIMONOTONE TYPE 427

3. Semimonotone set-valued maps In this section, we first prove the existence theorem for differential inclusion (1.1) when F is a semimonotone set-valued map on a finite dimensional space and then we generalize it to the infinite dimensional Hilbert spaces. Note that without loss of generality, we can assume that x(0) = 0.

n n Theorem 3.1. Let the Caratheodory set-valued map F : [0,T ]×R → Pc,v(R ) satisfy hypothesis (H2) and the differential equation u′ = h(t,u) with u(0) = 0 has a solution u(t) existing on [0,T ]. Then, the differential inclusion x′(t) ∈ F (t, x(t)), t ∈ [0,T ], (3.1)  x(0) = 0, has a classical solution. Proof. We are to show that there exists a function x ∈ C(I; Rn) such that t x(t) ∈ 0 F (s, x(s))ds for all t ∈ [0,T ]. To this end, we define the set-valued map ΛR on C(I; Rn) as t Rn 1 Λ(x)= {y : I → | y(t)= v(s)ds, ∀t ∈ [0,T ] for some v ∈ SF,x}. Z0 Since each Caratheodory map is measurable, by Kuratowski-Ryll Nardzewski selection theorem and hypothesis (H2), the map Λ has non-null values. We consider the closed convex subset K = {x ∈ C(I; Rn): x(0) = 0, kx(t)k≤ u(t) ∀t ∈ [0,T ]}, of the space C(I, Rn). First, we show that Λ : K → P (K). Indeed, for all 1 t ∈ [0,T ], x ∈ K and v ∈ SF,x we have t t t t v(s) ≤ kv(s)k ds ≤ h(s, kx(s)k)ds ≤ h(s,u(s))ds = u(t). Z0 Z0 Z0 Z0

∞ Next, it is easy to see that the values of Λ are convex. Let {yn}n=1 be a sequence in Λ(x) which is convergent to y in C(I, Rn). Then, there exists a ∞ 1 t sequence {vn}n=1 in SF,x such that yn(t)= 0 vn(s)ds for all t ∈ [0,T ]. Since t 7→ h(t,u(t)) is a continuous function and R

kvn(t)k≤kF (t, x(t))k≤ h(t,u(t)) ∞ for each t ∈ [0,T ] and every n ≥ 1, we observe that {vn}n=1 is in fact a se- quence of elements of the Hilbert space L2([0,T ]; Rn). So, it has a subsequence (we denote it by the same symbol) that is weakly convergent to a function n v ∈ L2 ([0,T ]; R ), say. Then, we conclude from Corollary V·3·14 of [14] that ∞ there is a sequence {wn}n=1 of convex combinations of {v1, v2,...} which is, in particular, convergent to v in the norm topology of the space L1 ([0,T ]; Rn). 1 ∞ Since F has convex values, wn ∈ SF,x. The sequence {wn}n=1 has a subse- quence (we denote it again by the same symbol) which converges to v a.e. on 428 HOSSEINABEDIANDRUHOLLAHJAHANIPUR

1 [0,T ]. Because the values of F are closed, we have v ∈ SF,x and the dominated convergence theorem gives t t y(t) = lim y (t) = lim v (s)ds = v(s)ds. →∞ n →∞ n n n Z0 Z0 Thus, Λ(x) is a closed set. Now, we prove that Λ is a closed map. Let sequences ∞ ∞ {xn}n=1 and {yn}n=1 in K converge to the continuous functions x and y in the n space C(I; R ), respectively and furthermore yn ∈ Λ(xn) for all n. By upper semicontinuity of F with respect to the second variable, for given ε> 0, large enough n and any t ∈ [0,T ], we have F (t, xn(t)) ⊆ F (t, x(t)) + εB. So, t t yn(t) ∈ F (s, xn(s))ds ⊆ F (s, x(s))ds + tεB. Z0 Z0 t Since ε was arbitrary, we obtain yn(t) ∈ 0 F (s, x(s))ds = Λ(x). This together with the result that Λ(x) is closed, impliesR that y(t) ∈ Λ(x) for all t ∈ [0,T ]. Finally, we show that Λ(K) is uniformly bounded and equicontinuous in the Banach space C(I; Rn). Uniform boundedness of Λ is a consequence of conti- nuity of the function u on [0,T ] and the following dominance relation which by hypothesis H2, is true for all x ∈ K and y ∈ Λ(x): T T sup ky(t)k≤ kF (s, x(s))kds ≤ h(s,u(s))ds = u(T ). t∈I Z0 Z0 Also, given y ∈ Λ(K), we can prove similarly that

ky(t1) − y(t2)k ≤ |u(t1) − u(t2)| for all t1,t2 ∈ [0,T ]. Thus, by the Arzela-Ascoli theorem, Λ(K) is a relatively compact subset of the space C(I; Rn). So, Λ is compact and it is also u.s.c by Proposition 2.1. Now, by Kakutani’s fixed point theorem [17], Λ has a fixed point in K which is a classical solution of the differential inclusion (3.1). 

In the above theorem, if we replaced the upper semicontinuity condition on F by lower semicontinuity, then according to Theorem 2.2, there is a single- valued Caratheodory function f : [0,T ] × Rn → Rn such that f(t, x) ∈ F (t, x). Then, Theorem 3.2 of [20], indicates that the ordinary differential equation x′(t)= f(t, x(t)), t ∈ [0,T ],  x(0) = 0, has a generalized solution on [0,T ]. Thus, the above differential inclusion has a generalized solution on [0,T ]. So we have the following theorem.

n n Theorem 3.2. Let the integrable set-valued map F : [0,T ]×R → Pc,v(R ) be l.s.c respect to the second variable and satisfy condition (H2). If the differential equation u′ = h(t,u) with u(0) = 0 has a solution u(t) existing on [0,T ], then the differential inclusion (3.1) has a generalized solution. NONLINEAR DIFFERENTIAL INCLUSIONS OF SEMIMONOTONE TYPE 429

Now, we generalize the previous theorem to infinite dimensional real sepa- rable Hilbert spaces. Denote by Cl0 (I; H) the space of all continuous functions x I H x L2 I H,M : → such that (0) = 0 equipped with sup-norm. Also, l0 ( ; ) stands for the Hilbert space of all measurable functions x : I → H such that T − e 2Mt kx(t)k2 dt < ∞, Z0 with the inner product T −2Mt hx, yiM = e hx(t),y(t)i dt, Z0 and the norm 1 T 2 −2Mt 2 kxkM = e kx(t)k dt! . Z0 Proposition 3.3. Let the Caratheodory multifunction F : [0,T ] × H → P (H) satisfy hypothesis (H2). Define the set-valued map G on Cl0 (I; H) as 1 Gx = {y : I → H | y ∈ SF,x, y(t) ∈ F (t, x(t)) for all t ∈ I}. Then G C I H P L2 I H,M is bounded on bounded subsets of (a) : l0 ( ; ) → l0 ( ; )

Cl0 (I; H).  (b) If F satisfies hypothesis (H1), then G is semimonotone with constant M. (c) If F satisfies hypothesis (H3), then G is hemicontinuous. Proof. We refer to [20] for the proof of parts (a) and (b). In particular, if

B ⊆ Cl0 (I; H) is a bounded set, then there exists a constant K > 0 such that 1 kGxk ≤ γ(T )K, where γ(T ) = √1 1 − e−2MT 2 . To prove (c), suppose M 2M ∞ that {xn}n=1 is a sequence which converges to xin Cl0 (I; H). Since F is ∞ hemicontinuous, if y ∈ G(x), then there is a sequence of functions {yn}n=1 such w that yn(t) ∈ F (t, xn(t)) for each n ≥ 1 and t ∈ [0,T ] and also, yn(t) → y(t) in H as n → ∞. According to hypothesis (H2), we have {y }∞ ⊆ L2 (I; H,M) n n=1 l0 y t y t ,z t n z L2 I H,M and h n( ) − ( ) ( )i → 0 as → ∞ for every ∈ l0 ( ; ). Moreover, the Cauchy-Schwarz inequality implies that |hyn(t) − y(t),z(t)i| ≤ K kz(t)k for some constant K > 0. Now, by the dominated convergence theorem, we have T −2Mt |hyn − y,ziM |≤ e |hyn(t) − y(t),z(t)i| dt → 0, Z0 w as n → ∞. Therefore, y → y in L2 (I; H,M) and the proof is complete.  n l0 Next, we generalize Lemma 1.2 of [9] for the set-valued hemicontinuous maps. It is needed to prove the main theorem of this section. 430 HOSSEINABEDIANDRUHOLLAHJAHANIPUR

Lemma 3.4. Suppose that D is dense subspace of Hilbert space H and the set-valued map F : D ⊆ H → P (H) satisfy hypothesis (H3). If u0 ∈ D, y0 ∈ H and M > 0 is a constant such that 2 (3.2) hy0 − y,u0 − ui≤ M ku0 − uk for all u ∈ D and y ∈ F (u), then y0 ∈ F (u0).

Proof. Assume that y0 ∈/ F (u0). Fix y ∈ F (u0). Since y−y0 6= 0 and D is dense 1 in H, we can find vy ∈ D such that hy − y0, vyi > 0. Setting un = u0 + n vy for each n ≥ 1, we have un ∈ D and by hypothesis H3, we can take yn ∈ F (un) w such that yn → y in H. By (3.2), we have for all n ∈ N that 1 kv k2 hy − y , v i≥−M y . n 0 n y n2 Therefore, kv k2 hy − y , v i ≥ hy − y , v i− M y . n y 0 y n Now, letting n → ∞, we obtain 0 < hy − y0, vyi ≤ 0. This is a contradiction. 

Our main theorem in this section extends Theorem 3.1 and 3.2 to differential inclusions on infinite dimensional real separable Hilbert spaces. Theorem 3.5. Let H be a real separable Hilbert space and the Caratheodory set-valued map F : [0,T ] × H → Pc,v(H) satisfy hypotheses (H1)–(H3). If the differential equation u′ = h(t,u) with u(0) = 0 has a solution u(t) existing on [0,T ], then the differential inclusion problem x′(t) ∈ F (t, x(t)), t ∈ [0,T ], (3.3)  x(0) = 0, has a unique classical solution. ∞ Proof. Take an orthonormal basis {en}n=1 for H and let Hn be the subspace ∞ of H generated by {e1,...,en}. Then, {Hn}n=1 is an increasing sequence of ∞ finite dimensional subspaces of H with the property ∪n=1Hn = H. If Pn is the orthogonal projection of H onto Hn, then each Pn induces a corresponding projection of L2 (I; H,M) onto subspace L2 (I; H ,M), we denote it again by l0 l0 n Pn: P u t P u t H , t I,u L2 I H,M . ( n )( )= n ( ) ∈ n ∀ ∈ ∈ l0 ( ; ) ∗ ∗ We have Pn = Pn → I as n → ∞ where Pn is the adjoint operator of P and I is the identity map on H. For each n ≥ 1, we define the set-valued map PnF : [0,T ] × Hn → Pc,v(Hn) by PnF (t, x) = {Pn(y) | y ∈ F (t, x)}. Theorem 1.2.8 of [21] yields that PnF is a Caratheodory map and since

kPnF (t, x)k = sup kPnyk≤kF (t, x)k≤ h(t, kxk), y∈F (t,x) NONLINEAR DIFFERENTIAL INCLUSIONS OF SEMIMONOTONE TYPE 431 it also satisfies hypothesis (H2). By Theorem 3.1, for each n ≥ 1, the differential inclusion x′(t) ∈ P F (t, x(t)), t ∈ [0,T ], (3.4) n  x(0) = 0, has a classical solution xn : [0,T ] → Hn. On the other hand, hypothesis (H1) holds for PnF . In fact, let xi ∈ Hn and yi ∈ PnF (t, xi) for i =1, 2. Then there ∗ exist zi ∈ F (t, xi), i =1, 2, such that yi = Pnzi. Since Pn = Pn , we get 2 hy2 − y1, x2 − x1i = hz2 − z1, x2 − x1i≤ M kx2 − x1k . ∞ Thus, just as the proof of Proposition 3.5 of [20], the sequence {xn}n=1 is bounded by a constant which depends only on F and T and we have T 2 (2M+1)T 2 kxnk∞ ≤ e kF (s, 0)k ds. Z0 ∞ Therefore, according to part (a) of Proposition 3.3, G({xn}1 ) is a bounded L2 I H,M L2 I H,M set in l0 ( ; ). Now, we define the linear operator Γ : l0 ( ; ) →

Cl0 (I; H) by t (Γy)(t)= y(s)ds. Z0 Since 1 T 2 −2Mt 2 k(Γy)(t)k≤ γ(T ) e ky(t)k dt! = γ(T ) kykM , Z0 1 where γ(T ) = √1 1 − e−2MT 2 , Γ is continuous. Consider the set-valued 2M map ΓPnGx = {Γy |y ∈ PnGx}from Cl0 (I; H) into Cl0 (I; Hn). Let xn be a solution of the differential inclusion (3.4), so that xn ∈ ΓPnGxn for all n ≥ 1. We can find y ∈ L2 (I; H,M) such that x = ΓP y . Since y ∈ Gx , n l0 n n n n n ∞ 2 the sequence {yn}n=1 is bounded in Ll (I; H,M) and it is possible to extract 0 ∞ a subsequence (we denote it again by {yn}n=1) which is weakly convergent w w in L2 (I; H,M) . Suppose that y → y as n → ∞. Then, x → Γy = x ∈ l0 n n

Cl0 (I; H). It is enough to show that y ∈ Gx. With the same method as in the proof of Theorem 3.6 of [20], one can show that 2 hw − y, x − uiM ≥−M kx − ukM , ∀u ∈ Cl0 (I; ,H), w ∈ Gu. Since, by Proposition 3.3, G is hemicontinuous, the result follows from the Lemma 3.4. Finally, we come to prove the uniqueness of solution. If x1 and x2 are two solutions of problem (3.3), then we have x2(t) − x1(t) = t 1 1 0 (v1(s) − v2(s))ds for some v1 ∈ SF,x1 and v2 ∈ SF,x2 and Lemma 3.4 of R[20] shows that t 2 2 x2(t) − x1(t) ≤ 2M x2(s) − x1(s) ds. Z 0 Thus, by the Gronwall inequality, x2(t) − x1(t) = 0 for all t ∈ [0,T ]. 

432 HOSSEINABEDIANDRUHOLLAHJAHANIPUR

Now, if F is integrable hemicontinuous in the previous theorem, then one can easy see that the set-valued map PnF : [0,T ] × H → Pc,v(Hn) is hemicon- tinuous. So, PnF is l.s.c with respect to the second variable, since weak and strong convergence are equivalent on Hn. Then, by Theorem 3.2 the differen- tial inclusion (3.4) has a generalized solution xn for all n. Therefore, with a similar proof as Theorem 3.5 we obtain the following theorem. Theorem 3.6. Let H be a real separable Hilbert space and the integrable set- valued map F : [0,T ] × H → Pc,v(H) satisfy hypotheses (H1)–(H3). If the differential equation u′ = h(t,u) with u(0) = 0 has a solution u(t) existing on [0,T ], then the differential inclusion (3.3) has a unique generalized solution.

4. Condensing set-valued maps In this section, we discuss the existence of a solution for differential inclusion (3.3) in the case that the set-valued map F is condensing with respect to a regular measure of noncompactness on a real separable Hilbert space. we get a global existence theorem that is the analogous results of the Theorems 5.2.1 of [21], under a continuous bounded. Our main result in this section is the following Theorem 4.1. Let H be a real separable Hilbert space. Suppose that the Caratheodory (integrable) set-valued map F : [0,T ] × H → Pc,v(H) satisfy hypotheses (H2) and there is a function k ∈ L1([0,T ]; [0, ∞)) such that for every countable pointwise bounded subset Ω of C(I; H), we have α(F (t, Ω(t))) ≤ k(t)α(Ω(t)), where α is the Hausdorff measure of noncompactness. If the differential equa- tion u′ = h(t,u) with u(0) = 0 has a solution u(t) existing on [0,T ], then the differential inclusion (3.3) has a classical (generalized) solution on [0,T ]. To prove the theorem, we need a very useful lemma the proof of which can be found in [21]. consider the set K ⊂ C(I,H) as K = {x ∈ C(I; H): x(0) = 0, kx(t)k≤ u(t) ∀t ∈ [0,T ]}. ∞ Lemma 4.2. Suppose that all conditions of Theorem 4.1 hold. Let {xn}n=1 be ∞ 1 a sequence in K which converges to x and {vn}n=1 be a sequence in L ([0,T ]; H) w 1 such that vn → v in L ([0,T ]; H). If for each n ≥ 1 and for a.e. t ∈ [0,T ], 1 t t vn(t) ∈ F (t, xn(t)), then v ∈ SF,x. Moreover, 0 vn(s)ds tends to 0 v(s)ds in the space C(I; H) R R Proof. Consider the K ⊆ C(I; H) as above and the set-valued map Λ in- troduced in the proof of Theorem 3.1. Since H is separable, Λ has non- null values and according to the hypotheses (H2), kΛ(x)(t)k ≤ u(t) for all (t, x) ∈ [0,T ] × K. First, we show that Λ is closed. To this end, let the se- ∞ ∞ quences {xn}n=1 and {yn}n=1 converge to x and y, respectively, in the space NONLINEAR DIFFERENTIAL INCLUSIONS OF SEMIMONOTONE TYPE 433

C(I; H) and yn ∈ Λ(xn) for all n ∈ N. So, there exists a sequence of functions ∞ 1 1 N {vn}n=1 in L ([0,T ]; H) such that vn ∈ SF,xn for each n ∈ and

t yn(t)= vn(s)ds, ∀t ∈ [0,T ]. Z0

We obtain from hypothesis (H2) that kvn(t)k≤kF (t, xn(t))k≤ h(t, kxn(t)k) ≤ ∞ h(t,u(t)). Thus, {vn}n=1 is integrably bounded and moreover since h is contin- ∞ uous, {vn(t)}n=1 is relatively weakly compact in H for all t ∈ [0,T ]. Therefore, w 1 we can assume that vn → v in L ([0,T ]; H) (see [5]). Now, by Lemma 4.2, 1 t v ∈ SF,x and yn(t) → 0 v(s)ds in C([0,T ]; H) as n → ∞. Consequently, we t R have y(t) = 0 v(s)ds. This proves that Λ is a closed set-valued map and a similar argumentR indicates that Λ(x) is also closed for every x ∈ K. Next, we ∞ show that Λ(x) is compact for all x ∈ K. Given any sequence {yn}n=1 ⊆ Λ(x), 1 t we can find for every n ≥ 1, a vn ∈ SF,x such that yn(t) = 0 vn(s)ds for all t ∈ [0,T ]. Again passing to a suitable subsequence, if necessary,R we may as- w n→∞ → 1 t → t sume that vn v in L (I; H). Thus, by Lemma 4.2, 0 vn(s)ds 0 v(s)ds in C([0,T ]; H). Therefore, Λ(x) is compact. Now, letR us assume thatR χ is a regular measure of noncompactness we defined at the end of Section 2. Assume ∞ that Ω ⊆ C(I; H) is bounded and χ(Ω) ≤ χ(Λ(Ω)). If {xn}n=1 is a sequence in −Lt ∞ −Lt ∞ Ω, then sup e α({xn(t)}n=1) ≤ sup e α({yn(t)}n=1), where yn ∈ Λ(xn). t∈I t∈I Therefore, we obtain

−Lt ∞ sup (e α({xn(t)}n=1)) t∈[0,T ] t −Lt ∞ ≤ sup (e α({ vn(s)ds}n=1)) t∈[0,T ] Z0 t −Lt −Lu ∞ Ls ≤ sup (e sup (e α({xn(u)}n=1) e k(s)ds)) t∈[0,T ] u∈[0,T ] Z0 t −Lu ∞ −L(t−s) ≤ sup (e α({xn(u)}n=1)) sup ( e k(s)ds). u∈[0,T ] t∈[0,T ] Z0

t −L(t−s) If we choose constant L in such a way that supt∈[0,T ] 0 e k(s)ds < ∞ ∞ 1, then we obtain αC ({xn}n=1) = 0. Note that sinceR {yn}n=1 is a rela- ∞ tively compact subset of C([0,T ]; H), then αC ({yn}n=1) = 0. Moreover, ∞ modC ({yn}n=1) = 0. Thus, χ(Λ(Ω)) = 0 and consequently, χ(Ω) = 0. This shows that the set-valued map Λ is χ- condensing. Therefore, according to The- orem 2.4, Λ has a fixed point which is a generalized solution of the differential inclusion (3.3).  434 HOSSEINABEDIANDRUHOLLAHJAHANIPUR

5. Mild solutions In this section, we come to prove the existence of mild solution for semilinear differential inclusion x′(t) − Ax(t) ∈ F (t, x(t)), t ∈ [0,T ], (5.1)  x(0) = ξ, with ξ ∈ H. Bya mild solution to (5.1), we mean a continuous function x : [0,T ] → H such that for every t ∈ [0,T ], t x(t)= S(t)ξ + S(t − s)v(s)ds, Z0 1 where v ∈ SF,x. If for a.e. t ∈ [0,T ], v(t) ∈ D(A) and the function t 7→ Av(t) is integrable, then for every ξ ∈ D(A), the mild solution is absolutely continuous and therefore, almost- everywhere differentiable. Theorem 5.1. Suppose that H is a separable Hilbert space and F : [0,T ]×H → Pc,v(H) is a integrable (Caratheodory) set-valued map satisfying hypotheses (H1)–(H3). If A : D(A) ⊆ H → H is the infinitesimal generator of a con- ′ traction C0 semigroup {S(t)}t≥0 and the differential equation u = h(t,u) with u(0) = 0 has a solution u(t) existing on [0,T ], then the semilinear differential inclusion (5.1) has a unique mild solution. Proof. By the Hille-Yosida theorem, we have for every n ∈ N that

− 1 (nI − A) 1 ≤ . n

−1 Define Rn = n(nI − A) and An = ARn. Since Anx = (nRn − nI)x for each x ∈ H, we have kAnk≤ 2n. Therefore, the bounded linear operator An is the infinitesimal generator of the uniformly continuous semigroup eAnt. Then, we define the set-valued map Fn : [0,T ]×H → Pc,v(H) by Fn(t, x)= Anx+F (t, x). We show that Fn satisfies hypotheses (H1)–(H3). Since by the Lumer-Philips theorem, A is dissipative and Rnx ∈ D(A) for every x ∈ H, we have 1 hA x, xi = hAR x, xi = hAR x, R xi− hAR x, AR xi n n n n n n n 1 ≤− kAR xk2 ≤ 0. n n

Therefore, An is also dissipative. Take x1, x2 ∈ H, t ∈ [0,T ] and y1 ∈ Fn(t, x1), y2 ∈ Fn(t, x2). Then, there are z1 ∈ F (t, x1) and z2 ∈ F (t, x2) such that yi = Anxi + zi for i =1, 2. So, we obtain

hy1 − y2, x1 − x2i = hAnx1 + z1 − Anx2 − z2, x1 − x2i

= hAn(x1 − x2), x1 − x2i + hz1 − z2, x1 − x2i 2 ≤ M kx1 − x2k . NONLINEAR DIFFERENTIAL INCLUSIONS OF SEMIMONOTONE TYPE 435

This implies that Fn is semimonotone with constant M. Define hn(t,s) = h(t,s)+2ns for every (t,s) ∈ [0,T ] × [0, ∞). We have

kFn(t, x)k = sup kAnx + yk≤kAnxk + kF (t, x)k y∈F (t,x)

≤ 2n kxk + h(t, kxk)= hn(t, kxk). ∞ This shows that Fn satisfies hypothesis (H2). Finally, suppose that {xm}1 is a sequence in H which is convergent to x. If y ∈ Fn(t, x), then there is a ∞ w sequence {zm}1 in H such that zm ∈ F (t, xm), zm → z = y − Anx ∈ F (t,y). w Now, assuming ym = zm + Anx, we have ym → y in H and ym ∈ Fn(t, xm). Consequently, for every n ∈ N, Fn is hemicontinuous. By hypothesis (H2), F takes the bounded sets in [0,T ] × H to the bounded subsets of H. Since An is a , Fn has this property also. Now, according to Theorem 3.5 (Theorem 3.5), there exists a unique generalized (classical) solution xn : [0,T ] → H for the differential inclusion ′ x (t) ∈ Fn(t, x(t)), t ∈ [0,T ],  x(0) = ξ.

Hence, xn is a strong solution of the integral equation t t x(t)= ξ + Anx(s)ds + v(s)ds, Z0 Z0 1 where v ∈ SF,x. The solution xn is of the form

t Ant An(t−s) xn(t)= e ξ + e v(s)ds, ∀t ∈ [0,T ]. Z0 1 N Here vn ∈ SF,xn for all n ∈ . Since D(An) = H, xn is a classical solution d and we have dt xn = Anxn + vn, xn(0) = ξ. By the Hille-Yosida theorem, we Ant know that S(t)x = limn→∞ e x for all x ∈ H and moreover, the convergence is uniform on [0,T ]. We conclude by Energy-Type inequality in [25] that, the ∞ sequence of approximate solutions {xn}n=1 is bounded by a constant which depends only on M and T ; in other words, we have

2 kxn(t)k ≤ γT,M , ∀t ∈ [0,T ], n ∈ N, where T 2M+1 2 2 γT,M = e kξk + h (t, 0)dt! . Z0

1 On the other hand, since vn ∈ SF,xn , we have kvn(t)k≤ h(t,γT,M ) for a.e. t ∈ ∞ [0,T ]. Therefore, we can assume that {vn}n=1 is weakly convergent to v in 1 ∞ L (0,T ; H) and so, there exists a sequence {wn}n=1 of convex combinations of 1 {vn} such that wn → v in L (0,T ; H) as n → ∞ . We extract a subsequence 436 HOSSEINABEDIANDRUHOLLAHJAHANIPUR of it (denoted by the same symbol) which is a.e. convergent to v. Since, F has 1 close convex values, wn ∈ SF,xn . Define t Ant An(t−s) yn(t)= e ξ + e wn(s)ds Z0 and t x(t)= S(t)ξ + S(t − s)v(s)ds, Z0 for all t ∈ [0,T ]. For every z ∈ H we obtain

hz,yn(t) − x(t)i t

Ant An(t−s) = z,e ξ − S(t)ξ + z, e wn(s) − S(t − s)v(s)ds * Z + 0 t Ant An(t−s) = z,e ξ − S(t)ξ + z,e wn(s) − S(t − s)v(s) ds. Z 0 D E w By the dominated convergence theorem, we conclude that yn(t) → x(t) as n → w ∞ for all t ∈ [0,T ] and since the sequence is integrably bounded, yn → x in L1 ([0,T ]; H). Eventually, we use again Lemma 3.4 and conclude from the semimonotonicity of G that v ∈ Gx. 

6. Example In this section, we give an example to illustrate the application of the results obtained in this paper. Assume that Ω is an open bounded subset of R2 with boundary ∂Ω of class C2. Consider the second order differential inclusion ∂2u ∂u ∈ F (t, x, u, ), ∂t2 ∂t with the boundary condition u(t, x) = 0, for each (t, x) ∈ [0,T ]×∂Ω and initial ∂u 1 2 conditions ∂t (0, x) = ψ(x),u(0, x) = ϕ(x). Here, ϕ ∈ H0 (Ω), ψ ∈ L (Ω) and 2 the Caratheodory multivalued map F : [0,T ] × Ω × R → Pb,c,v(R) satisfies the following hypotheses:

(I) There is a constant M > 0 such that for every t ∈ [0,T ], u1,u2 ∈ R 2 and v1, v2 ∈ L (Ω), we have

2 (ψ1(x) − ψ2(x))(v1(x) − v2(x))dx ≤ M kv1 − v2kL2(Ω) , ZΩ

in which the functions ψ1, ψ2 :Ω → R are such that

ψ1(x) ∈ F (t,x,u1, v1(x))

and ψ2(x) ∈ F (t,x,u2, v2(x)) for all x ∈ Ω. NONLINEAR DIFFERENTIAL INCLUSIONS OF SEMIMONOTONE TYPE 437

(II) There is an increasing continuous function k(t) such that kF (t,x,u,v)k2 ≤ k(t) |u|2 + |v|2   for all (t,x,u,v) ∈ [0,T ] × Ω × R2. ∞ 1 (III) For any sequence {un}n=1 in H0 (Ω) which is convergent to u and se- ∞ 2 quence {vn}n=1 in L (Ω) which is convergent to v, if there exists a function z :Ω → R such that z(x) ∈ F (t,x,u(x), v(x)) for all t ∈ [0,T ], w then one can find a sequence of functions zn :Ω → R such that zn → z 2 in L (Ω) and moreover, zn(x) ∈ F (t,x,un(x), vn(x)). 1 2 Now, we consider the Hilbert space H = H0 (Ω)×L (Ω) with the inner product ′ ′ h(u1, v1), (u2, v2)iH = u1(x)u2(x)dx + u1(x)u2(x)dx + v1(x)v2(x)dx. ZΩ ZΩ ZΩ We define the multivalued map G on [0,T ]×H by G(t,z)(x)={0}×F (t,x,z(x)). Then, G is a Caratheodory multivalued map with closed and convex values. It is easy to see that G satisfies hypotheses (H1)–(H3). Therefore, defining ∂u z(t)x = (u(t, x), ∂t (t, x)), we can reduce the above second order differential inclusion to the equivalent first order differential inclusion z′(t) ∈ G(t,z(t)),  z(0) = (ϕ, ψ). 1 Hence, by Theorem 3.5, it has a unique mild solution u ∈ C [0,T ]; H0 (Ω) ∂u 2 such that ∂t ∈ C [0,T ]; L (Ω) almost everywhere on [0,T ].   References

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Hossein Abedi Department of Mathematical Sciences University of Kashan 87317-51167, Ravand Road, Kashan, Iran E-mail address: [email protected]

Ruhollah Jahanipur Department of Mathematical Sciences University of Kashan 87317-51167, Ravand Road, Kashan, Iran E-mail address: [email protected]